General Optimal Stopping without Time Consistency

In this paper, we propose a new framework for solving a general dynamic optimal stopping problem without time consistency. A sophisticated solution is proposed and is well-defined for any time setting with general flows of objectives. A backward iteration is proposed to find the solution. The iteration works with an additional condition, which holds in interesting cases including the time inconsistency arising from non-exponential discounting. Even if the iteration does not work, the equilibrium solution can still be studied by a forward definition.

💡 Research Summary

The paper addresses dynamic optimal stopping problems in which the decision maker’s preferences are not time‑consistent, a situation that arises in many economic and financial contexts (e.g., hyperbolic discounting, moving profit targets, health‑related quitting decisions). Classical optimal stopping theory relies on the dynamic programming principle, which assumes that a plan optimal at time zero remains optimal at every later date. When this principle fails, the usual notion of an optimal stopping time becomes meaningless, and a new equilibrium concept is required.

The authors first review the “backward induction solution” (BIS) that works in finite discrete‑time settings. BIS defines a sequence of stopping times (\tau^B_t) by comparing the payoff from stopping immediately at (t) with the payoff from following the previously computed optimal plan at (t+1). This construction yields a unique “sophisticated” stopping rule in the finite‑horizon, discrete‑time case, but it does not extend naturally to continuous‑time or infinite‑horizon problems.

To overcome this limitation, the paper introduces a general framework that works for any time set (\mathcal{T}) (finite or infinite, discrete, continuous, or hybrid). The key objects are:

1. The mapping (F) – for any stopping time (\rho),
   \